A couple of points to the very fine comments already. Your choice really depends on how you prioritize your use for this lens.

A previous poster explained that IS is less important than fast/good autofocus; when it comes to photographing children, I totally agree. If this is important to you then the 70-300L becomes a stronger choice because it has newer, more advanced AF electronics that work better with your 5D3. Roger at Lensrentals has documented this in abundance; look for his blog entries on autofocus. With the 70-300L and your 5D3, you can shoot at higher ISO/higher shutter speed and have better autofocus response. It is also a very compact lens that is very versatile for landscapes, etc. While some may quibble with its optical performance compared to other canon L lenses, I find it to be excellent and functionally equivalent. I own a 135 F2.0L and 200 F2.8L and find the images from the 70-300L are equally usable in terms of sharpness and overall image quality.

If portraits are really your thing, then consider a 135 f2.0L the extra lens speed and handling will be a better fit for portrait situations. If you are looking for longer reach, then consider a 200 F2.8L - this lens is superb.

Probably the best advice you received is to rent one or more of these and try it for yourself. Lensrentals makes it easy with good selection and fair prices (no I don't work for them but have rented from them)...The tradeoffs in the answer to your question are subtle enough that all of our recommendations on this forum will only take you so far...you have to experience them for yourself.

I'm not going to tell you what lens to buy because everyone's use case and budget is different. But I'm tired of blanket statements about how terrible certain lenses are when it's just not true. Here are two examples of pictures taken with the 70-300mm IS (non-L) lens. They are not "quite bad".Cheers,DJD

You are unlikely to get a better value for money lens than the 70-200 f/4 (non-IS). It was my first L lens and since then I've purchased only Ls ... It's addictive.

+1

I have the 70-200 2.8IS and I have borrowed the F4 version for comparison. The L series glass is great and I highly suggest it. If you need the additional reach, you could pick up the 1.4 Teleconverter when you have the budget and get out to 280 @ f5 which is pretty close to 300 f5.6.

I have the 70-200 2.8IS and I have borrowed the F4 version for comparison. The L series glass is great and I highly suggest it. If you need the additional reach, you could pick up the 1.4 Teleconverter when you have the budget and get out to 280 @ f5 which is pretty close to 300 f5.6.

do you need the zoom or the reach?considered just picking up a 200 f2.8L

no IS either but it is a stop faster and is smaller and less conspicuous

Well, for shooting portraits and candids, the reach may be really useful. Likewise the IS in a lower light scenario. I went from the 70-200 f/4L IS to the 200 f/2.8 because it was faster and I shot a lot of action (and used it mostly at 200mm anyway). So, IS didn't help me, but the extra stop does.

Considering you already have a 50mm and 85mm prime, then I'd say a tele-zoom is probably a nice accompaniment to that. The 70-200's are both great, and the suggestion for the 70-300L is good too. If you want it now, the non-IS lens is a great value, and your 5dIII will allow you to get away wiht pretty high ISO's. If you can wait, both the L IS lenses are pretty stunning